On Completeness of Ultra Real Number System and Strictly Positive Measure

Abstract

Just about two decades ago Abraham Robinson introduced Ultra real number system R* in Mathematics [1]. Real number system R is inadequate incase of strictly positive measure, since an infinitesimal (Ï R) is needed in such situations. Prasad has quoted two cases where real numbers fail: Measurement of horn angles and weight of functions [3].

In case of Real numbers we know that R is Cauchy complete, order-complete and also Dedekind-complete and these three forms of completeness are equivalent in R[4]; which is an Archimedean ordered field. But since R* is not Archimedean, validity of these three forms of completeness needs to be examined. It is also desirable to examine the adequacy of R* for situations demanding strictly positive measure. In this present paper it is proposed to discuss these two cases.